Can I use previously rendered frames as texture in Cycles?
The answer to this is basically the same as Is it possible to use the output of a Renderlayer in the material nodes of another RenderLayer?, except you must render the first frame repeatedly until the replicated images of the render are smaller than can be seen (this of course depends on the resolution)

I've thought about this, and as far as I can tell, there is really no true way around it using compositing.

The reason for this is that this is a feedback loop; 
The rendered result depends on the lighting, which is affected by the texture of the rendered result, which is affected by the lighting, which is affected by the rendered result, etc.

The only thing you can do is try and make the initial render a reasonably close approximation of how it will look after enough iterations have been done to make it appear infinite (i.e. emitting bright pink doesn't help). This way it will look "good enough" in fewer iterations.

Using the setup in my other answer, rendering a single frame will write a file via file output node which will then be read by an image sequence node in material nodes the next time the scene is rendered. So all you have to do is press F12 repeatedly..





To automate pressing F12, you could render an "animation" where nothing changes for as many frames as you want iterations.